On simplifying
`(sin^3A+sin3A)/(sinA)+(cos^3A-cos3A)/(cosA)` we get

Options

A. sin3A
B. cot3A
C. sinA+cosA
D. 3

Answer

To simplify the expression \((\sin^3 A + \sin 3A)/\sin A + (\cos^3 A - \cos 3A)/\cos A\), we will follow these steps:

### Step 1: Use the trigonometric identities for \(\sin 3A\) and \(\cos 3A\)

We know the following identities:
- \(\sin 3A = 3 \sin A - 4 \sin^3 A\)
- \(\cos 3A = 4 \cos^3 A - 3 \cos A\)

### Step 2: Substitute the identities into the expression

Substituting these identities into the expression gives us:

\[ \frac{\sin^3 A + (3 \sin A - 4 \sin^3 A)}{\sin A} + \frac{\cos^3 A - (4 \cos^3 A - 3 \cos A)}{\cos A} \]

### Step 3: Simplify the numerator of each fraction

Now, simplify the numerators:

1. For the sine part:
\[ \sin^3 A + 3 \sin A - 4 \sin^3 A = -3 \sin^3 A + 3 \sin A = 3 \sin A - 3 \sin^3 A \]

2. For the cosine part:
\[ \cos^3 A - (4 \cos^3 A - 3 \cos A) = \cos^3 A - 4 \cos^3 A + 3 \cos A = -3 \cos^3 A + 3 \cos A \]

### Step 4: Rewrite the expression

Now, we can rewrite the expression as:

\[ \frac{3 \sin A - 3 \sin^3 A}{\sin A} + \frac{-3 \cos^3 A + 3 \cos A}{\cos A} \]

### Step 5: Factor out the common terms

Factoring out the common terms gives us:

\[ 3 \left(1 - \sin^2 A\right) + 3 \left(1 - \cos^2 A\right) \]

### Step 6: Use the Pythagorean identity

Using the Pythagorean identity \(1 - \sin^2 A = \cos^2 A\) and \(1 - \cos^2 A = \sin^2 A\), we can rewrite the expression:

\[ 3 \cos^2 A + 3 \sin^2 A \]

### Step 7: Combine the terms

Since \(\cos^2 A + \sin^2 A = 1\), we have:

\[ 3(\cos^2 A + \sin^2 A) = 3 \cdot 1 = 3 \]

### Final Result

Thus, the simplified expression is:

\[ \boxed{3} \]

---